We give an improved explicit construction of highlyunbalanced bipartite expander graphs with expansion arbitrarilyclose to the degree (which is polylogarithmic in the number ofvertices). Both the degree and the number of right-hand verticesare polynomially close to optimal, whereas the previousconstructions of Ta-Shma, Umans, and Zuckerman (STOC `01) requiredat least one of these to be quasipolynomial in the optimal. Ourexpanders have a short and self-contained description and analysis,based on the ideas underlying the recent list-decodableerror-correcting codes of Parvaresh and Vardy (FOCS `05).

Our expanders can be interpreted as near-optimal ``randomnesscondensers,'' that reduce the task of extracting randomness fromsources of arbitrary min-entropy rate to extracting randomness fromsources of min-entropy rate arbitrarily close to 1, which is a mucheasier task. Using this connection, we obtain a new, self-containedconstruction of randomness extractors that is optimal up to constantfactors, while being much simpler than the previous construction of Luet al. (STOC `03) and improving upon it when the error parameter issmall (e.g. 1/poly(n)).

Paper:

TR06-134 | 18th October 2006 00:00

Extractors and condensers from univariate polynomials

We give new constructions of randomness extractors and lossless condensers that are optimal to within constant factors in both the seed length and the output length. For extractors, this matches the parameters of the current best known construction [LRVW03]; for lossless condensers, the previous best constructions achieved optimality to within a constant factor in one parameter only at the expense of a polynomial loss in the other.

Our constructions are based on the Parvaresh-Vardy codes [PV05], and our proof technique is inspired by the list-decoding algorithm for those codes. The main object we construct is a condenser that loses only the entropy of its seed plus one bit, while condensing to entropy rate $1 - \alpha$ for any desired constant $\alpha > 0$. This construction is simple to describe, and has a short and completely self-contained analysis. Our other results only require, in addition, standard uses of randomness-efficient hash functions (to obtain a lossless condenser) or expander walks (to obtain an extractor).

Our techniques also show for the first time that a natural construction based on univariate polynomials (i.e., Reed-Solomon codes) yields a condenser that retains a $1 - \alpha$ fraction of the source min-entropy, for any desired constant $\alpha > 0$, while condensing to constant entropy rate and using a seed length that is optimal to within constant factors.